F-mixing and weak disjointness

A topological system (X,f) is $\text{F}$-transitive if for each pair of opene subsets U and V of X, $\text{N}{}_{\mathrm{f}}\text{(U,V)={n∈}\text{Z}{}_{\mathrm{+}}\text{:}\text{f}{}^{\mathrm{n}}\text{U∩V≠∅}∈}\text{F}$, where $\text{F}$ is a collection of subsets of $\text{Z}{}_{\mathrm{+}}$ which is hereditary upward. (X,f) is $\text{F}$-mixing if (X×X,f×f) is $\text{F}$-transitive. In this paper $\text{F}$-mixing systems are characterized in terms of the chaoticity of the systems. Moreover, weak disjointness is studied via family. We will give conditions such that a dual theorem of the Weiss–Akin–Glasner theorem holds. Examples with this dual theorem fails for some “good” families are obtained.